Induction from an abelian normal subgroup with a free character orbit

Let $N◃G$ be abelian and let a linear character $\theta$ of $N$ have stabilizer exactly $N$ under conjugation by $G$. Then ${\mathrm{Ind}}_N^G\theta$ is irreducible of degree $[G:N]$. Its value is zero outside $N$, while on $n\in N$ it is the sum of the characters in the $G$-orbit of $\theta$.

For $G=C_{11}⋊C_5$ with faithful action, the abelianization $G/C_{11}\cong C_5$ supplies five linear characters. The two orbits of size five on the nontrivial characters of $C_{11}$ induce two irreducible characters of degree five. Their squared degrees satisfyso these seven characters are all the irreducible characters of $G$.